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Tricky heat calculation from the 1850s
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[QUOTE="TSny, post: 6199754, member: 229090"] Clausius’ goal is to prove that heat is not a state variable, contrary to the opinion of some scientists of that time period. In order to follow his argument, it is very important to understand the notation that Clausius is using. I also think that it’s important to understand the “Mathematical Introduction” on pages 1 -13. The notation ## \frac{dQ}{dv}## associated with a state, ##A##, of the system can be defined operationally as follows. Imagine the system starting in state A. Add a very small amount of heat ##dQ## to the system [I]while keeping the temperature ##t## of the system constant[/I]. Let ##dv## be the corresponding change in volume (assumed to be nonzero). The meaning of ## \frac{dQ}{dv}## is simply the ratio of the two quantities ##dQ## and ##dv##. This definition does not require us to think of ##\frac{dQ}{dv}## as a partial derivative of some function ##Q(v, t)##. Clausius’s goal is to show that such a function ##Q(v,t)## does not exist! The ratio ##\frac{dQ}{dv}## was defined above for the state ##A## since we took state ##A## as the initial state when adding the heat ##dQ##. So, it might have been clearer to write the notation as ##\frac{dQ}{dv}|_A##. Clearly, we can carry out the definition for any initial state, ##S##, of the system to get ##\frac{dQ}{dv}|_S##. Thus, there is a value of ##\frac{dQ}{dv}|_S## at each state ##S##. But, a state is determined by values of ##v## and ##t##. So, you can think of ## \frac{dQ}{dv}## as defining some function ##F(v,t)##. Note that ##F## [U]is[/U] a state variable! It has a definite value for each state of the system. In going isothermally from state ##A## to state##B## in your diagram, the heat added is ##\left ( \frac{dQ}{dv}|_A \right) dv##, where ##dv## is the change in volume when going from ##A## to ##B##. Likewise, if you were to start at ##D## and go isothermally to ##C##, the heat added would be ##\left ( \frac{dQ}{dv}|_D \right) d’v##, where ##d’v## is the change in volume when going from ##D## to ##C##. (The heat removed when going the other way from ##C## to ##D## is just the negative of this.) The heat added in going from ##D## to ##C## can be expressed as ##F(v_D, t_D) d’v##. Since ##D## is close to ##A##, we can write ##F(v_D, t_D)## in terms of ##F(v_A, t_A)## to sufficient accuracy as ##F(v_D, t_D) = F(v_A, t_A) + \frac {\partial F}{\partial v}|_A \delta v + \frac {\partial F}{\partial t}|_A (-dt)##. Note that the volume change in going from ##A## to ##D## is ##\delta v## and the temperature change in going from ##A## to ##D## is ##(-dt)## according to Clausius’ definitions of symbols. So, the heat that would be added in going from ##D## to ##C## can be expressed as ## F(v_D, t_D) d’v =\left[ F(v_A, t_A) + \frac {\partial F}{\partial v}|_A \delta v - \frac {\partial F}{\partial t}|_A dt\right] d’v##. This is essentially the same as Clausius’ expression that you were asking about: [ATTACH type="full" alt="245872"]245872[/ATTACH] Clausius uses the notation ##\frac{dQ}{dv}## for ##\frac{dQ}{dv}|_A = F(v_A, t_A)##. [/QUOTE]
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Tricky heat calculation from the 1850s
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